The conditional cumulative distribution function in single functional index model

ABSTRACT

We consider the estimation of the conditional cumulative distribution function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional median estimate. Also, the choice of the functional index via the cross-validation procedure is also discussed but not attacked.     

Recursive kernel estimate of the conditional quantile for functional ergodic data

ABSTRACT

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variable Y given a random variable (rv) X taking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators.     

Varying-Coefficient Functional Linear Regression Models

This article considers a generalization of the functional linear regression in which an additional real variable influences smoothly the functional coefficient. We thus define a varying-coefficient regression model for functional data. We propose two estimators based, respectively, on conditional functional principal regression and on local penalized regression splines and prove their pointwise consistency. We check, with the prediction one day ahead of ozone concentration in the city of Toulouse, the ability of such nonlinear functional approaches to produce competitive estimations.     

Genetic Algorithm in the Wavelet Domain for Large p Small n Regression

Many areas of statistical modeling are plagued by the “curse of dimensionality,” in which there are more variables than observations. This is especially true when developing functional regression models where the independent dataset is some type of spectral decomposition, such as data from near-infrared spectroscopy. While we could develop a very complex model by simply taking enough samples (such that n > p), this could prove impossible or prohibitively expensive. In addition, a regression model developed like this could turn out to be highly inefficient, as spectral data usually exhibit high multicollinearity. In this article, we propose a two-part algorithm for selecting an effective and efficient functional regression model. Our algorithm begins by evaluating a subset of discrete wavelet transformations, allowing for variation in both wavelet and filter number. Next, we perform an intermediate processing step to remove variables with low correlation to the response data. Finally, we use the genetic algorithm to perform a stochastic search through the subset regression model space, driven by an information-theoretic objective function. We allow our algorithm to develop the regression model for each response variable independently, so as to optimally model each variable. We demonstrate our method on the familiar biscuit dough dataset, which has been used in a similar context by several researchers. Our results demonstrate both the flexibility and the power of our algorithm. For each response variable, a different subset model is selected, and different wavelet transformations are used. The models developed by our algorithm show an improvement, as measured by lower mean error, over results in the published literature.     

Asymptotic Results of a Nonparametric Conditional Quantile Estimator for Functional Time Series

We consider the estimation of the conditional quantile function when the covariates take values in some abstract function space. The main goal of this article is to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional quantile under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications and particular cases are studied. This approach can be applied in time series analysis to the prediction and building of confidence bands. We illustrate our methodology with El Niño data.     

Robust group-Lasso for functional regression model

In this article, we consider the problem of selecting functional variables using the L1 regularization in a functional linear regression model with a scalar response and functional predictors, in the presence of outliers. Since the LASSO is a special case of the penalized least-square regression with L1 penalty function, it suffers from the heavy-tailed errors and/or outliers in data. Recently, Least Absolute Deviation (LAD) and the LASSO methods have been combined (the LAD-LASSO regression method) to carry out robust parameter estimation and variable selection simultaneously for a multiple linear regression model. However, variable selection of the functional predictors based on LASSO fails since multiple parameters exist for a functional predictor. Therefore, group LASSO is used for selecting functional predictors since group LASSO selects grouped variables rather than individual variables. In this study, we propose a robust functional predictor selection method, the LAD-group LASSO, for a functional linear regression model with a scalar response and functional predictors. We illustrate the performance of the LAD-group LASSO on both simulated and real data.     

Functional Analysis of Variance,Discriminant Analysis,and Clustering in a Manifold of Elastic Curves

We develop functional data analysis techniques using the differential geometry of a manifold of smooth elastic functions on an interval in which the functions are represented by a log-speed function and an angle function. The manifold's geometry provides a method for computing a sample mean function and principal components on tangent spaces. Using tangent principal component analysis, we estimate probability models for functional data and apply them to functional analysis of variance, discriminant analysis, and clustering. We demonstrate these tasks using a collection of growth curves from children from ages 1–18.     

Functional variable selection via Gram–Schmidt orthogonalization for multiple functional linear regression

ABSTRACT

Functional linear model is of great practical importance, as exemplified by applications in high-throughput studies such as meteorological and biomedical research. In this paper, we propose a new functional variable selection procedure, called functional variable selection via Gram–Schmidt (FGS) orthogonalization, for a functional linear model with a scalar response and multiple functional predictors. Instead of the regularization methods, FGS takes into account the similarity between the functional predictors in a data-driven way and utilizes the technique of Gram–Schmidt orthogonalization to remove the irrelevant predictors. FGS can successfully discriminate between the relevant and the irrelevant functional predictors to achieve a high true positive ratio without including many irrelevant predictors, and yield explainable models, which offers a new perspective for the variable selection method in the functional linear model. Simulation studies are carried out to evaluate the finite sample performance of the proposed method, and also a weather data set is analysed.     

Sparse group lasso for multiclass functional logistic regression models

Sparsity-inducing penalties are useful tools for variable selection and are also effective for regression problems where the data are functions. We consider the problem of selecting not only variables but also decision boundaries in multiclass logistic regression models for functional data, using sparse regularization. The parameters of the functional logistic regression model are estimated in the framework of the penalized likelihood method with the sparse group lasso-type penalty, and then tuning parameters for the model are selected using the model selection criterion. The effectiveness of the proposed method is investigated through simulation studies and the analysis of a gene expression data set.     

P-spline Estimation of Functional Classification Methods for Improving the Quality in the Food Industry

The aim of this article is to improve the quality of cookies production by classifying them as good or bad from the curves of resistance of dough observed during the kneading process. As the predictor variable is functional, functional classification methodologies such as functional logit regression and functional discriminant analysis are considered. A P-spline approximation of the sample curves is proposed to improve the classification ability of these models and to suitably estimate the relationship between the quality of cookies and the resistance of dough. Inference results on the functional parameters and related odds ratios are obtained using the asymptotic normality of the maximum likelihood estimators under the classical regularity conditions. Finally, the classification results are compared with alternative functional data analysis approaches such as componentwise classification on the logit regression model.     

Rate of uniform consistency for nonparametric estimates with functional variables

In this paper we investigate nonparametric estimation of some functionals of the conditional distribution of a scalar response variable Y given a random variable X taking values in a semi-metric space. These functionals include the regression function, the conditional cumulative distribution, the conditional density and some other ones. The literature on nonparametric functional statistics is only concerning pointwise consistency results, and our main aim is to prove the uniform almost complete convergence (with rate) of the kernel estimators of these nonparametric models. Unlike in standard multivariate cases, the gap between pointwise and uniform results is not immediate. So, suitable topological considerations are needed, implying changes in the rates of convergence which are quantified by entropy considerations. These theoretical uniform consistency results are (or will be) key tools for many further developments in functional data analysis.     

Regression model for surrogate data in high dimensional statistics

Abstract

This paper deals with the problem of estimating the regression of a surrogated scalar response variable given a functional random one. We construct an estimator of the regression operator by using, in addition to the available (true) response data, a surrogate data. We then establish some asymptotic properties of the constructed estimator in terms of the almost-complete and the quadratic mean convergences. Notice that the obtained results generalize a part of the results obtained in the finite dimensional framework. Finally, an illustration on the applicability of our results on both simulated data and a real dataset was realized. We have thus shown the superiority of our estimator on classical estimators when we are lacking complete data.     

Local linear estimate of the point at high risk: Spatial functional data case

Abstract

The purpose of the present paper is to investigate by the local linear method a nonparametric estimator of the point at high risk of scalar response variable given a functional variable when the observations are spatially dependent. The main goal is to establish the almost complete convergence with rate of this estimator under some general conditions. A practical example on the climatological data shows the usefulness of our theoretical study.     

Asymptotic normality of the local linear estimation of the conditional density for functional time-series data

This article focuses on the conditional density of a scalar response variable given a random variable taking values in a semimetric space. The local linear estimators of the conditional density and its derivative are considered. It is assumed that the observations form a stationary α-mixing sequence. Under some regularity conditions, the joint asymptotic normality of the estimators of the conditional density and its derivative is established. The result confirms the prospect in Rachdi et al. (2014 Rachdi, M., A. Laksaci, J. Demongeot, A. Abdali, and F. Madani. 2014. Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data. Computational Statistics and Data Analysis 73 :53–68.[Crossref], [Web of Science ®] , [Google Scholar]) and can be applied in time-series analysis to make predictions and build confidence intervals. The finite-sample behavior of the estimator is investigated by simulations as well.     

Function-on-function regression for two-dimensional functional data

ABSTRACT

We present methods for modeling and estimation of a concurrent functional regression when the predictors and responses are two-dimensional functional datasets. The implementations use spline basis functions and model fitting is based on smoothing penalties and mixed model estimation. The proposed methods are implemented in available statistical software, allow the construction of confidence intervals for the bivariate model parameters, and can be applied to completely or sparsely sampled responses. Methods are tested to data in simulations and they show favorable results in practice. The usefulness of the methods is illustrated in an application to environmental data.     

Bootstrap in functional linear regression

We have considered the functional linear model with scalar response and functional explanatory variable. One of the most popular methodologies for estimating the model parameter is based on functional principal components analysis (FPCA). In recent literature, weak convergence for a wide class of FPCA-type estimates has been proved, and consequently asymptotic confidence sets can be built. In this paper, we have proposed an alternative approach in order to obtain pointwise confidence intervals by means of a bootstrap procedure, for which we have obtained its asymptotic validity. Besides, a simulation study allows us to compare the practical behaviour of asymptotic and bootstrap confidence intervals in terms of coverage rates for different sample sizes.     

Wavelet modeling of functional random effects with application to human vision data

In modern statistical practice, it is increasingly common to observe a set of curves or images, often measured with noise, and to use these as the basis of analysis (functional data analysis). We consider a functional data model consisting of measurement error and functional random effects motivated by data from a study of human vision. By transforming the data into the wavelet domain we are able to exploit the expected sparse representation of the underlying function and the mechanism generating the random effects. We propose simple fitting procedures and illustrate the methods on the vision data.     

Bayesian covariance estimation and inference in latent Gaussian process models

This paper describes inference methods for functional data under the assumption that the functional data of interest are smooth latent functions, characterized by a Gaussian process, which have been observed with noise over a finite set of time points. The methods we propose are completely specified in a Bayesian environment that allows for all inferences to be performed through a simple Gibbs sampler. Our main focus is in estimating and describing uncertainty in the covariance function. However, these models also encompass functional data estimation, functional regression where the predictors are latent functions, and an automatic approach to smoothing parameter selection. Furthermore, these models require minimal assumptions on the data structure as the time points for observations do not need to be equally spaced, the number and placement of observations are allowed to vary among functions, and special treatment is not required when the number of functional observations is less than the dimensionality of those observations. We illustrate the effectiveness of these models in estimating latent functional data, capturing variation in the functional covariance estimate, and in selecting appropriate smoothing parameters in both a simulation study and a regression analysis of medfly fertility data.     

Inferences for a Partially Varying Coefficient Model With Endogenous Regressors

In this article, we propose a new class of semiparametric instrumental variable models with partially varying coefficients, in which the structural function has a partially linear form and the impact of endogenous structural variables can vary over different levels of some exogenous variables. We propose a three-step estimation procedure to estimate both functional and constant coefficients. The consistency and asymptotic normality of these proposed estimators are established. Moreover, a generalized F-test is developed to test whether the functional coefficients are of particular parametric forms with some underlying economic intuitions, and furthermore, the limiting distribution of the proposed generalized F-test statistic under the null hypothesis is established. Finally, we illustrate the finite sample performance of our approach with simulations and two real data examples in economics.